ArticlePDF Available

Abstract and Figures

Our recent works discuss the meaning of an arbitrary-order SIR model. We claim that arbitrary-order derivatives can be obtained through special power-laws in the infectivity and removal functions. This work intends to summarize previous ideas and show new results on a meaningful model constructed with Mittag-Leffler functions. We emphasize the tricky idea to deal with equilibria, the nonlocality of the model and the non-intuitive behavior near the lower terminal.
Content may be subject to copyright.
Vol. 51, 2542 ©2022
Some remarks on an arbitrary-order SIR
model constructed with Mittag-Leffler
Noemi Zeraick Monteiro 1and Sandro Rodrigues
Mazorche 1
1Universidade Federal de Juiz de Fora, Juiz de Fora, Brazil
Abstract. Our recent works discuss the meaning of an arbitrary-
order SIR model. We claim that arbitrary-order derivatives can be
obtained through special power-laws in the infectivity and removal
functions. This work intends to summarize previous ideas and show
new results on a meaningful model constructed with Mittag-Leffler
functions. We emphasize the tricky idea to deal with equilibria, the
nonlocality of the model and the non-intuitive behavior near the
lower terminal.
Keywords: Fractional Calculus. Epidemiological model. Mittag-
Leffler functions. Nonlocality. Non-intuitive behaviors.
2020 Mathematics Subject Classification: 12A34, 67B89.
1 Introduction
“Mathematics is biology’s next microscope, only better; biology is math-
ematics’ next physics, only better” [1]. The quote by biomathematician
The first author is supported by the Coordination for the Improvement of Higher
Education Personnel (CAPES) - Brazil - Financing Code 001; e-mail of the correspond-
ing author:
26 N. Zeraick Monteiro and S. R. Mazorche
Joel E. Cohen portrays the current development of mathematical biology,
a branch that holds the attention of several researchers and has been on
the agenda every day during the current COVID-19 pandemic. The wide
use of mathematical models in situations related to biology does not ex-
haust their study. Contrariwise, some mathematical tools need a revision
to be used properly.
A great tool for problem modeling is Arbitrary-Order Calculus, known
as Fractional Calculus. In the most used definitions, there is the possi-
bility of explicitly considering the dependence of previous stages of the
phenomenon studied, through the nonlocality of the operators. This is
generally related to the “memory effect” [2]. The recent explosion of pub-
lications in Fractional Calculus highlights its immense applicability in nu-
merous areas (see, for instance, the data collected in [2]). However, the
basis is still not unified and coherent.
Compartmental models, for example, have been widely studied with
arbitrary orders. Generally, they are obtained by replacing an integer
derivative with an arbitrary-order one. Throughout our research, we pub-
lish some works about meaning difficulty, loss of properties and the lack
of the construction of fractional SIR type models ([3], [4], [5]). In this
context, we study a model proposed by Angstmann, Henry and McGann
[6] in which the arbitrary-order derivatives are obtained by construction,
considering Mittag-Leffler functions and generalizing the infectivity and
remotion functions. We seek to extend some analytical and numerical
results of the model in [4], [7], [8], [9].
Here, one of the aims is, after some preliminary presentations, dis-
cuss the idea around how to find equilibria in the proposed model, since
equating the right side of the system to zero is no longer a viable strategy.
The discussed strategy is somewhat trickier, using Laplace transform tech-
niques. After that, the main propose deals with two points that were not
discussed yet: as one would expect, the model is nonlocal and, moreover,
presents a non-intuitive behavior in the lower terminal.
Some remarks on an arbitrary-order SIR model 27
2 A brief note about nonlocality and the Frac-
tional Calculus
The impetus theory, studied by names as Leonardo Da Vinci, deals
with the concept of “impression”. According to Da Vinci, the impression is
maintained during a certain time in its sensitive object. This impression
(memory) characterized by a long time produces more lasting effects, while
a short memory produces effects that occur in shorter times [10]. Although
this reasoning was lost in the context of the Newtonian physics, in the last
century the quantum theory and the modern string theory allowed a revival
of nonlocal theories. Within that context, the Fractional Calculus can be
seen as a continuation or a sublimation of nonlocal concepts [11].
Broadly speaking, the Fractional Calculus, parallel to the delay differ-
ential equations, has been shown to be a very useful tool in capturing the
dynamics of the physical process of several scientific objects. Probably, it
was born in 1695, when l’Hôpital asked Leibniz about the meaning of a
derivative of order 1/2. Over the subsequent centuries, important advances
were made by Liouville, Riemann, Grünwald, Caputo, and many others.
However, it was only after the first International Conference on Fractional
Calculus and Applications, in 1974, that the number of researchers in Frac-
tional Calculus showed great growth. The reader may refer to the reference
[12] for a detailed chronology of publications in Fractional Calculus until
2019, as well as for general results.
Nonlocal operators can be constructed in different ways, depending on
the bias worked on. Here, before introducing the arbitrary-order integral,
we recall the concept of the integer-order integral, sometimes called the
multiple or iterated integral:
Definition 2.1 (Integer-order iterated integral).The integral of order
nNis defined by the expression
Inf(t) = Zt
· · · Ztn2
f(tn)dtndtn1· · · dt3dt2dt1.(2.1)
By definition, I0f(t) = f(t).
28 N. Zeraick Monteiro and S. R. Mazorche
The next result, using the Laplace convolution, is a starting point for
the generalization of the concept of an integral of order n. For that, we
Definition 2.2 (Gel’fand-Shilov function).Let αR,α > 0. The
Gel’fand-Shilov function is defined as
ϕα(t) =
Γ(α)if t0,
0if t < 0,
where Γrepresents the gamma function.
Theorem 2.3. Let nN,0< t < and f(t)be an integrable function.
Inf(t) = ϕn(t) f(t) = Zt
(n1)! f(τ)dτ, (2.3)
where indicates convolution [13].
Thus, it is to be expected that the definition of an integral of arbitrary
order αis given by Iαf(t) = ϕα(t) f(t). Below, we consider [a, b]a finite
real interval, and αa real number such that 0n1< α < n, with n
Definition 2.4 (Riemann-Liouville integral in finite intervals).The Riemann-
Liouville integral of an arbitrary order αis set to t[a, b]by
a+f(t) = 1
(tθ)α1f(θ)dθ. (2.4)
After introducing the arbitrary-order integral, it is natural to search
for the definition of the corresponding derivative. There are several defi-
nitions of these kind of derivatives, each one constructed with a particular
viewpoint. In this work, we use the Riemann-Liouville’s one:
Definition 2.5 (Riemann-Liouville derivative in finite intervals).The
Riemann-Liouville derivative of an arbitrary order αis set to t[a, b]
a+f(t) = Dn[Inα
a+f(t)] = 1
(tθ)nα1f(θ) , (2.5)
Some remarks on an arbitrary-order SIR model 29
with Dnrepresenting the integer-order derivative.
Finally, we present the Mittag-Leffler functions with one, two, and
three parameters. The classic Mittag-Leffler function, due to its impor-
tance in several arbitrary-order differential equations, was nicknamed the
“queen of special functions” of the Fractional Calculus. Its importance
for Fractional Calculus is analogous to the significance of the exponential
function for classical Calculus. We present the following definition [12]:
Definition 2.6 (Mittag-Leffler function with one, two, and three param-
eters).Let zbe a complex number, and three parameters α, β complex,
and ρreal, such that Re(α)>0, Re(β)>0, ρ > 0. We define the Mittag-
Leffler function with three parameters through the power series
α,β (z) =
Γ(αk +β)
where (ρ)kis the Pochhammer symbol, defined by (ρ)k= Γ(ρ+k)/Γ(ρ).
The three-parameter Mittag-Leffler function is also called Prabhakar
function. Particularly, when ρ= 1, we have (ρ)k=k!. In this case,
the definition recovers the two-parameter Mittag–Leffler function, denoted
simply by E1
α,β(t) = Eα,β (t). When ρ=β= 1, we obtain the classic
Mittag-Leffler function, denoted by E1
α,1(t) = Eα,1(t) = Eα(t). Finally, we
recover the exponential function when α=β=ρ= 1.
3 The model
We present in [4] a physical derivation following the steps of Angst-
mann, Henry & McGann [6], which use the probabilistic language of Con-
tinuous Time Random Walks (CTRW), and Mittag-Leffler functions. As
we can see with more detail in the references, the first idea is to consider
an individual infected since the time t. If there are S(t)susceptible in
time t, this infected person has a probability S(t)/N that his contact is
susceptible, considering the population homogeneous. Therefore, in the
period of tto t+ T, the expected number of new infections per infected
30 N. Zeraick Monteiro and S. R. Mazorche
individual is given by σ(t, t)S(t)∆T/N. The transmission rate per infec-
tious individual σ(t, t)depends on both the age of the infection, tt,
and the present time, t. The probability that an individual infected at the
moment tis still infected at the moment tis given by the survival function
Φ(t, t). Therefore, the flux of individuals to the Icompartment at a time
tis recursively given by
q+(I, t) = Zt
σ(t, t)S(t)
NΦ(t, t)q+(I, t)dt.(3.1)
To deal with the individuals infected at the time 0, we consider the
time in which each individual has become infected. This is given by the
function i(t,0) which represents the number of individuals who are still
infectious at time 0and who were originally infected at some point earlier
t<0. Then, q+(I, t) = i(t,0)/Φ(0, t)for t<0. For simplicity, we
consider i(t, 0) = i0δ(t), where δ(t)is the Dirac delta function. So,
q+(I, t) = Zt
σ(t, t)S(t)
NΦ(t, t)q+(I, t)dt+i0σ(t, 0)S(t)
NΦ(t, 0).(3.2)
As said, the infection rate σ(t, t)is assumed to be a function of both
the current time (due, for example, to containment measures), having an
extrinsic infectivity, ω, and the age of infection, tt, having an intrinsic
infectivity, ρ. So, we can write
σ(t, t) = ω(t)ρ(tt).(3.3)
Assuming that the natural death and the removal of an infected indi-
vidual are independent processes, we can write the survival function as
Φ(t, t) = ϕ(tt)θ(t, t),(3.4)
where ϕ(tt)is the probability that an individual infected since thas not
yet recovered or been killed by the disease at time t. Also, θ(t, t)is the
probability that an infected individual since thas not yet died of natural
death (that is, independent of the disease) until time t. The θfunction is
given by θ(t, t) = eRt
tγ(u)du,where γis the death rate.
Some remarks on an arbitrary-order SIR model 31
We define infectivity and recovery memory kernels
KI(t) = L1L{ρ(t)ϕ(t)}
L{ϕ(t)}, KR(t) = L1L{ψ(t)}
where ψ(t) = (t)/dt. This ψhas an important relationship with the
continuous random variable Xthat provides the time of removal of the
individual from the infectious compartment. The cumulative distribution
of X, namely Fdefined by F(t) = P(Xt), is such that F(t) = 1 ϕ(t).
Therefore, the probability density function of Xis ψ(t) = (t)/dt. We
can state the set of equations for the SIR model in a similar manner to
that written originally by Kermack and McKendrick [14]:
dt =γ(t)Nω(t)S(t)
Nθ(t, 0) Zt
dt =ω(t)S(t)
Nθ(t, 0) Zt
θ(t,0) dtθ(t, 0) Zt
θ(t,0) dtγ(t)I(t),
dt =θ(t, 0) Zt
where we consider the same rate γ(t)of natural mortality in each compart-
ment, with the birth rate equal to that. The population remains constant.
We choose ψ(t)and ρ(t)using Mittag-Leffler functions, in order to
generalize the exponential distribution of the random variable Xand allow
a variable intrinsic infectivity:
ϕ(t) = Eα,1t
τα, ρ(t) = 1
τβEα,β t
Using Laplace transform techniques, the Riemann-Liouville derivatives
arise along the construction and the SIR model, with 1βα > 0, is
given by
dt =γ(t)Nω(t)S(t)θ(t, 0)
Nτ βD1βI(t)
θ(t, 0)γ(t)S(t),(3.10)
dt =ω(t)S(t)θ(t, 0)
Nτ βD1βI(t)
θ(t, 0) θ(t, 0)
θ(t, 0) γ(t)I(t),(3.11)
dt =θ(t, 0)
θ(t, 0)γ(t)R(t),(3.12)
32 N. Zeraick Monteiro and S. R. Mazorche
Notice that, if α=β= 1, and γ(t)γ , ω(t)ωare considered
constant, we get the simple integer-order SIR model with constant coef-
ficients. Moreover, the cumulative distribution of Xis a Mittag-Leffler
distribution F(t;α, τ )=1Eα((t/τ)α). If α=β= 1, we have an ex-
ponential distribution and the expectation (first moment) of the random
variable Xexists, with τbeing exactly the average recovery time. When
α < 1, we do not have finite expectation.
Remark 3.1. In epidemics such as COVID-19, reports exhibit the asym-
metry of each infectious wave: “while COVID-19 accelerates very fast, it
decelerates much more slowly. In other words, the way down is much
slower than the way up” [15]. In addition, scientists suggest that infected
people are most infectious immediately before they develop symptoms and
at the onset of them (e.g. [16]). The two factors presented, that is, the
asymmetry of the data, with a heavy right tail effect, and the decrease in
infectivity over time since infection, are captured by the arbitrary-order
model presented. As started in [4], [17], [18], we have been working on
applications of the model to COVID-19.
3.1 Equilibrium
Here, we analyze the equilibrium point (S, I, R)such that lim
t→∞(S, I, R) =
(S, I, R), where the limit is taken coordinate by coordinate. We con-
sider γ(t)γconstant, so θ(t, 0) = eγt. Taking the limit t , the
model reduces to:
0 = γN lim
t→∞ ω(t)S(t)eγt
Nτ βD1β(I(t)eγ t)γS,(3.13)
0 = lim
t→∞ ω(t)S(t)eγt
Nτ βD1β(I(t)eγ t)eγ t
ταD1α(I(t)eγt )γI ,(3.14)
0 = lim
t→∞ eγt
ταD1α(I(t)eγt )γR.(3.15)
To calculate the limits of the form lim
t→∞ eγt D1α(I(t)eγt ), considering
γ > 0, we follow [19] and take the Laplace Transform:
L{eγt D1α(I(t)eγt )}= (s+γ)1αL{I}.(3.16)
Some remarks on an arbitrary-order SIR model 33
Using a Taylor series expansion, we get
(s+γ)1αL{I}=L{I}(γ1α+ (1 α)γαs+O(s2)).(3.17)
As the Laplace Transform is a linear operator, we can invert term by
term, obtaining
eγt D1α(I(t)eγt ) = γ1αI(t) + (1 α)γαdI
dt +L1(O(s2)).(3.18)
As we consider lim
t→∞ I(t) = I, we have
t→∞ dI/dt = lim
t→∞ L1(O(s2)) = 0.(3.19)
So, it follows that
t→∞ eγt D1α(I(t)eγt ) = γ1αI.(3.20)
Substituting these results into Eq. (3.13)-(3.15) and assuming that
t→∞ ω(t) = ω(possibly ω= 0), we are left with
0 = γN ωS
Nτ βγ1βIγS,(3.21)
0 = ωS
Nτ βγ1βI1
ταγ1αIγI ,(3.22)
0 = 1
These equations make it possible to obtain a disease-free state:
S=N, I = 0, R= 0,(3.24)
and, in the case where ω>0, we also obtain an endemic state:
S=((τ γ)βα+ (τ γ )β)N
ω, I=N(τ γ )α
1+(τ γ)αN(τ γ )β
ω, R=N
1+(τ γ)αN(τ γ )βα
When ω(t)ω, we get ω=ωand recover the endemic state of the
original article [6]. We observe that the endemic state makes physical sense
only if we can have I>0and R>0, that is, if
ω>(τγ)βα+ (τγ)β.(3.26)
34 N. Zeraick Monteiro and S. R. Mazorche
We have thus proven that, if there are asymptotically stable equilibria
in the case γ > 0, then they are given by Eq. (3.24)-(3.25). However,
we have not really proved that these states are asymptotically stable equi-
libria. We expect the disease-free state to be an asymptotically stable
equilibrium when ω<(τγ)βα+ (τ γ)β, while the endemic state must be
asymptotically stable if ω>(τγ)βα+ (τ γ)β.
For the case without vital dynamics, that is, γ= 0, if α=βand ω(t)
ω, we can write dS/dR =(ωS(t)/Nτ), as in the original model, also
discussed in [4]. Thus, the equilibrium point is the same as in the original
case, as we state in [7]. For the case with γ > 0, there are difficulties to
analyze formally the stability. Some advances were also reported in [7].
4 Main remarks
There are several numerical methods that can be applied to arbitrary-
order derivatives. For now, we build a numerical L1-scheme [20] to dis-
cretize the model described in the last Section. The time interval [a, t]
is discretized as a=t0< t1<· · · < tn=t, where the time steps
Ti=ti+1 ti, for i {0,· · · , n 1}, have the same size T. Consid-
ering α(0,1],we perform the following discretization for the Riemann-
Liouville derivative:
Γ(α+ 1) j1
f(tk)[(jk+ 1)α2(jk)α+ (jk1)α] + f(tj),(4.1)
where ti=iT+t0for all i {0,1,· · · , n}. It is important to state that
the integer-order case is obtained by taking α= 1.
4.1 Nonlocality
We illustrate that the model (3.10)-(3.12) is nonlocal, as expected. For
this, we consider N= 106and initial conditions S(0) = N1,I(0) =
1,R(0) = 0. At time t= 90, the numerical solution gives S(90) =
205890,I(90) = 267840, and R(90) = 526270. We now consider this
initial condition and run the model again.
Some remarks on an arbitrary-order SIR model 35
The Figure 4.1 illustrates the change of the solution, where the dashed
line corresponds to the solution from the time t= 90. In Figure 4.2,
we have the equivalent trajectories for a maximum time T= 3000. The
equilibrium is maintained.
0 50 100 150 200 250 300
t in days
S (blue), I (red), R(black)
Parameters: !=3; ==10; .=0.01
,=0.7; -=0.9
Figure 4.1: Change in the solution.
Parameters: !=3; ==10; .=0.01
,=0.7; -=0.9
Figure 4.2: Change in the trajectory.
Remark 4.1. In the integer-order SIR model, the epidemiological mean-
ingful parameters define the epidemic independently of time. Given an
initial condition (S(0), I(0), R(0)), and the solution (S(t), I (t), R(t)) of
the classic SIR model, let t>0: if we start at time t, with initial condi-
tion (S(t), I(t), R(t)), the solution will continue to be (S(t), I(t), R(t))
for t > t. This property of autonomous dynamical systems is called
invariance of solutions [21]. However, this is not valid here, making it
difficult to correctly choose initial conditions: what about the past prior
to the considered initial point? In this model, to adjust to the infection,
the parameters depend on the time series prior to the starting point. In
other words, different pasts will lead to different solutions in the future.
Starting a modeling in the second month of an epidemic, for example, does
not lead to the same result as when we start on the first day. It is worth
to mention that, to obtain the Equation (3.2), we used the Dirac delta
function to indicate that the disease does not exist before the time 0.
The nonlocality also implies an unexpected behavior of the reproduc-
tion number at time t. It can be understood as the expected number
of individuals that are infected by an infectious individual since t, with
36 N. Zeraick Monteiro and S. R. Mazorche
the basic reproduction number being the average number of secondary
infections that occur when an infectious individual is introduced into a
completely susceptible population [22]. In the constructed model, it is not
natural to define a formula that provides the reproduction number. Then,
in [6] the authors propose a construction for the basic reproduction num-
ber through an integral. Here, we extend the proposal of the reproduction
number for any time t. Thus, initially, we consider ω(t)constant and the
(t) = S(t)ω
Nτ β
This analysis is important, since for this model we do not have param-
eter tables, nor even prior application to any specific infectious disease.
As we can see here, in classical epidemiological models, the peak occurs
when (t)=1, but this does not occur in models of arbitrary orders or
with time-dependent parameters.
We consider the example below, in Figure 4.3, where the value of (t)
given by Eq. (4.3) for the peak point tis peak 0.9078 <1. If we
start modeling at a time before the peak, for which peak <(t)<1,
the epidemic will decrease even faster, but after a small rise. Note that
α=β. This will be discussed later. Also, is also possible to have no more
peaks if we start in a previous point, as we can see in Figure 4.4. In both
examples, we consider N= 106and the initial condition is (N1,1,0).
We also can observe peak >1. In some of these cases, we can take an
earlier day, where (t)>peak >1, and the disease starts decreasing even
faster, as illustrated in Figure 4.5. In other words, “the point where the
epidemic must start to decrease” is really not the peak point anymore! This
indicates a difficulty in understanding the 0defined in (4.2): although
0>1is the condition of endemic equilibrium viability given in (3.25),
0S(0)/N > 1does not necessarily indicate that the epidemic will or will
not occur, that is, whether or not Iwill grow.
Some remarks on an arbitrary-order SIR model 37
56 58 60 62 64 66 68 70
t in days
Parameters: !=2; ==4; .=0.01
,=0.5; -=0.5
Figure 4.3: Case (p)<1(α=β).
100 105 110 115 120 125 130 135
t in days
Parameters: !=2; ==7; .=0.01
,=0.5; -=0.6
Figure 4.4: Case (p)<1.
In this same Figure 4.5, we consider different starting points along
the originally constructed Icurve with initial condition (N1,1,0). So,
it illustrates the evolution of the Icompartment if we start modeling at
different starting points along the same curve. The nonlocal behavior of
the model is surprising and we see that, in this example, the Icompartment
only rises if we take a starting point much earlier than the peak. Here, we
also start to consider the proposal that we presented in [7] for a S-variable
reproduction number:
S(t) = Z
Nτ βω(t)eγ(tt)(tt)β1Eα,β (tt)
ταdt. (4.4)
In this proposal, the basic reproduction number is given by
Nτ βω(t)eγ ttβ1Eα,β t
ταdt. (4.5)
In the Figure table 4.6, we display the 0S(0)/N and the S
for each curve. In classical models, the disease declines since the beginning
if and only if 0S(0)/N < 1. If 0S(0)/N > 1, it must grow for a while.
As we can see, this does not happen with this model.
However, the equilibrium Igiven in (3.25) is the same for any initial
condition. So, broadly speaking, two simulations of the same disease with
different starting points lead to different solutions, but to the same end,
what is really important to study, for example, how many people will be
infected in total. Here we pointed out that, as will be explored in next
38 N. Zeraick Monteiro and S. R. Mazorche
25 30 35 40 45 50 55 60
t in days
Parameters: !=5; ==9; .=0.01
,=0.5; -=0.9
Figure 4.5: Modifying the start point. Figure 4.6: 0and S
0 100 200 300 400 500
t in days
Parameters: !=5; ==9; .=0.01
,=0.5; -=0.9
Figure 4.7: The equilibrium remains the same.
works, the Svariable reproduction number seems to provide something
closer to what we expect, in terms of plausible biological arguments.
4.2 Behavior near lower terminal
Finally, we exhibit an unusual feature observed in the initial instants
of the model simulations. One can easily see in the Figure 4.5 that the
Some remarks on an arbitrary-order SIR model 39
blue and purple curves have a small depression before a rise. This type of
behavior does not appear in the integer-order model.
In fact, this can be explained given the asymptotic behavior of the
Riemann-Liouville derivatives near the lower terminal [23]:
Γ(1 α)(ta)α,(ta+).(4.6)
So, in the model (3.10)-(3.12), if β > α, we have dI/dt < 0for tsmall
enough, as illustrated in the Figures 4.8-4.9.
0 0.2 0.4 0.6 0.8 1
t in days
Parameters: !=3; ==10; .=0.01
,=0.2:0.1:0.9; -=0.9
Figure 4.8: Lower terminal - Ex.1.
0 0.1 0.2 0.3 0.4
t in days
Parameters: !=3; ==10; .=0.01
,=0.4; -=0.4:0.1:0.9
Figure 4.9: Lower terminal - Ex.2.
When α=β, this behavior is not observed. For instance, we can see
in the pink curve of the Figure 4.3 that the behavior in the lower terminal
is the traditional movement of rising to a peak and decreasing after that.
Here, we also emphasize that the model assumes that there was no
disease before time 0, and then, at that point, an impulse caused the
disease to start. So, the historic is theoretically considered completely,
avoiding dubiousness about the starting point. However, the solution is not
C1, as it is not derivable at the origin. These remarks aim to strengthening
our intuition about the model and provide insights about its application.
5 Final considerations
So far, we have not been able to find a physical structure that allows
us to change the order of the original derivatives, even if the units are
40 N. Zeraick Monteiro and S. R. Mazorche
adjusted and the new orders are the same in all compartments. Therefore,
we seek the possibility of physically modeling a system, with a formalism
similar to that of Kermack and McKendrick, the creators of the SIR model.
In this sense, we have been discussing the almost unknown model of [6].
Here, we propagate the technique to work with equilibria in this type
of model and present important considerations, not previously explored:
the model is nonlocal, presents a difficult regarding the supposed relation
between the course of the epidemic and the reproduction number and, fi-
nally, there is an unexpected behavior at the initial point. Particularly,
start the simulation of an epidemic in its beginning or be able to say about
the past is a trick question. We aim that the deeper study of the equilib-
rium points and trajectories are fundamental predicting important features
of the model’s application. By other hand, the mathematics of the Frac-
tional Calculus’ models is still a black box of surprises that the assembling
between analytical and numerical studies can help to investigate.
Future works intend to close some results not fully explored, such as the
stability of equilibrium points in the general case and the use of an extrinsic
infectivity arising from Mittag-Leffler functions. The proposal of a variable
extrinsic infectivity was first illustrated in [4] and made it possible to fine-
tune the COVID-19 data, which leads us to new opportunities.
This study was financed in part by the Coordenação de Aperfeiçoamento de
Pessoal de Nível Superior Brasil (CAPES) Finance Code 001. We also thank
the team of the National Meeting of Mathematical Analysis and Applications
(ENAMA 2021), for the invitation to submit.
[1] COHEN, J. E. Mathematics is biology’s next microscope, only better; biology
is mathematics’ next physics, only better. PLoS biology, Public Library of
Science San Francisco, USA, v. 2, n. 12, p. e439, 2004.
Some remarks on an arbitrary-order SIR model 41
TARASOV, V. E. Trends, directions for further research, and some open prob-
lems of Fractional Calculus. Nonlinear Dynamics, Springer, p. 1–26, 2022.
[3] MAZORCHE, S. R.; MONTEIRO, N. Z. Modelos epidemiológicos fra-
cionários: o que se perde, o que se ganha, o que se transforma? In: Proceeding
Series of the Brazilian Society of Computational and Applied Mathematics.
Online: CNMAC, 2021.
[4] MONTEIRO, N. Z.; MAZORCHE, S. R. Fractional derivatives applied to
epidemiology. Trends in Computational and Applied Mathematics, p. 157–177,
[5] MONTEIRO, N. Z.; MAZORCHE, S. R. O Cálculo Fracionário e alguns de
seus comportamentos incomuns. In: I Reunião Mineira de Matemática. Online:
RMM, 2021.
[6] ANGSTMANN, C. N.; HENRY, B. I.; MCGANN, A. V. A fractional-order
infectivity and recovery SIR model. Fractal and Fractional, Multidisciplinary
Digital Publishing Institute, v. 1, n. 1, p. 11, 2017.
[7] MONTEIRO, N. Z.; MAZORCHE, S. R. Analysis and application of a frac-
tional SIR model constructed with Mittag- Leffler distribution. In: Proceedings
of the XLII Ibero-Latin-American Congress on Computational Methods in En-
gineering. Online: CILAMCE, 2021.
[8] MONTEIRO, N. Z.; MAZORCHE, S. R. Estudo de um modelo SIR fra-
cionário construído com distribuição de Mittag-Leffler. In: Proceeding Series
of the Brazilian Society of Computational and Applied Mathematics. Online:
CNMAC, 2021.
[9] MONTEIRO, N. Z.; MAZORCHE, S. R. Numerical study of the parameters
in an arbitrary order SIR model built with Mittag-Leffler distribution. C.Q.D.
Revista Eletrônica Paulista de Matemática, 2022. [to appear].
[10] MAGAZÙ, S.; COLETTA, N.; MIGLIARDO, F. Leonardo da Vinci: Cause,
effect, linearity, and memory. Journal of Advanced Research, Elsevier, v. 14,
p. 113–122, 2018.
[11] HERRMANN, R. Fractional Calculus: An Introduction For Physicists. Gi-
gaHedron, Germany: World Scientific, 2014.
42 N. Zeraick Monteiro and S. R. Mazorche
[12] OLIVEIRA, E. Capelas de. Solved Exercises in Fractional Calculus. Switzer-
land: Springer, 2019.
[13] CAMARGO, R. F.; OLIVEIRA, E. Capelas de. Cálculo fracionário. São
Paulo: Livraria da Fısica, 2015.
[14] KERMACK, W. O.; MCKENDRICK, A. G. Contributions to the mathe-
matical theory of epidemics–i. 1927. Bulletin of Mathematical Biology, v. 53,
n. 1-2, p. 33–55, 1991.
[15] ORGANIZATION, W. H. Director-general’s opening remarks at the media
briefing on COVID-19. World Health Organization, 2020.
[16] HART, W. S.; MAINI, P. K.; THOMPSON, R. N. High infectiousness im-
mediately before COVID-19 symptom onset highlights the importance of con-
tinued contact tracing. Elife, eLife Sciences Publications Limited, v. 10, p.
e65534, 2021.
[17] VICENTE, H. T.; MONTEIRO, N. Z.; MAZORCHE, S. R. Calibragem
do modelo epidemiológico SIR aplicado à COVID-19. In: Proceeding Series
of the Brazilian Society of Computational and Applied Mathematics. Online:
CNMAC, 2021.
[18] MONTEIRO, N. Z.; MAZORCHE, S. R. Application of a fractional SIR
model built with Mittag-Leffler distribution. In: Conference: Mathematical
Congress of the Americas. Online: MCA, 2021.
[19] ANGSTMANN, C.; HENRY, B.; MCGANN, A. A fractional order recov-
ery SIR model from a stochastic process. Bulletin of Mathematical Biology,
Springer, v. 78, n. 3, p. 468–499, 2016.
[20] OLDHAM, K.; SPANIER, J. The Fractional Calculus theory and applica-
tions of differentiation and integration to arbitrary order. New York: Elsevier,
[21] DOERING, C. I.; LOPES, A. O. Equações diferenciais ordinárias. Rio de
Janeiro: IMPA, 2008.
[22] HETHCOTE, H. W. The mathematics of infectious diseases. SIAM review,
SIAM, v. 42, n. 4, p. 599–653, 2000.
[23] PODLUBNY, I. Fractional differential equations: an introduction to frac-
tional derivatives, fractional differential equations, to methods of their solution
and some of their applications. San Diego: Elsevier, 1998.
Full-text available
A presentation of the latest results obtained in studies of Fractional Models by the team formed by Sandro R. Mazorche, Noemi Z. Monteiro and Matheus T. Mendonça. A brief historical presentation of Fractional Calculus with an emphasis on the turn of the 19th to the 20th century, where the figure of Magnüs G. Mittag-Leffler is the center of the initial discussion. Then a presentation of the Harmonic Oscillator, where a comparison is made: Fractional model x Integer model. In the third part, the SIR model is presented, and two fractional formulations are discussed: first, the model described changing the integer derivative by the Caputo derivative, where we will see solution behavior, lost of monotonicity and a correction in Barbalat's lemma for the fractional case. Then, we will see the description of the SIR model through a construction involving the derivative of Riemann-Liouville, where the central idea of the construction is presented, as well as the calculation of the asymptotic equilibrium points and stability results. Ending with a story involving Mittag-Leffler, Volterra and Fredholm. The presentation is in the link:
Full-text available Barbalat’s Lemma is a mathematical result that can lead to the solution of many asymptotic stability problems. On the other hand, Fractional Calculus has been widely used in mathematical modeling, mainly due to its potential to make explicit the dependence of previous stages through nonlocal operators. In this work, we present a fractional Barbalat’s Lemma and its proof, as proposed in [31]. The proof is analyzed in order to show an imprecision. In fact, for orders 0<α<1, we are not able to get the supreme limit of the integrand. Then, a counterexample and a corrected version of the lemma are presented, according to [9]. The objective of this work is to draw attention to the potential and limitations of a fractional Barbalat’s Lemma, given its wide use in recent articles. In a fractional SIR model, we exhibit the constraint of the result by introducing a non-periodic relapse. So, the supreme limit could not be verified. Also in this context, we provide a general discussion of the classical Calculus’ properties that are not inherited if we change the integer orders to fractional ones.
Full-text available
We previously discussed the construction of an arbitrary order SIR model with physical meaning. We believe that arbitrary order derivatives can be obtained through Mittag-Leffler based laws in the infectivity and removal functions. Here we are interested in obtaining information about the influence of the parameters in the studied model, with the aim of promoting its use and strengthening the biological interpretation. Keywords: Arbitrary order derivatives. SIR model. Mittag-Leffler. Parameter analysis
Full-text available
The area of fractional calculus (FC) has been fast developing and is presently being applied in all scientific fields. Therefore, it is of key relevance to assess the present state of development and to foresee, if possible, the future evolution, or, at least, the challenges identified in the scope of advanced research works. This paper gives a vision about the directions for further research as well as some open problems of FC. A number of topics in mathematics, numerical algorithms and physics are analyzed, giving a systematic perspective for future research.
Full-text available
O trabalho utilizou os dados liberados pelo governo do Brasil para calibrar os parâmetros β e γ, do modelo SIR, ajustando a curva dos infectados do modelo, I_m, o mais próximo possível a da evolução real do infectados, I_r. Para tal, utilizamos o método dos mínimos quadrados com restri¸ção, onde acoplamos o modelo SIR discretizado ao algoritmo FDIPA [2] -Feasible Direction Interior Point Algorithm.
Conference Paper
Full-text available
We have discussed previously the construction of an arbitrary-order SIR model with physical meaning. We believe that arbitrary-order derivatives can be obtained through potential laws in the infectivity and removal functions. This work intends to complete previous discussions, showing new results in a model with Mittag-Leffler distribution, with an emphasis on equilibrium points and reproduction numbers. We also discuss our prior application to COVID-19, now from the current perspective. VIDEO PRESENTATION AVAILABLE IN
Full-text available
We seek investigate the use of fractional derivatives, both analytically and through simulations. We present some models and perform investigations about them, starting with the classic model and the basic definitions to discuss difficulties in constructing a non-artificial fractional model. Also, we analyze the COVID-19 pandemic using a fractional epidemiological SIR model carefully constructed and present numerical results using MATLAB. Keywords: SIR model, Fractional Derivatives, COVID-19.
Full-text available
Background: Understanding changes in infectiousness during SARS-COV-2 infections is critical to assess the effectiveness of public health measures such as contact tracing. Methods: Here, we develop a novel mechanistic approach to infer the infectiousness profile of SARS-COV-2 infected individuals using data from known infector-infectee pairs. We compare estimates of key epidemiological quantities generated using our mechanistic method with analogous estimates generated using previous approaches. Results: The mechanistic method provides an improved fit to data from SARS-CoV-2 infector-infectee pairs compared to commonly used approaches. Our best-fitting model indicates a high proportion of presymptomatic transmissions, with many transmissions occurring shortly before the infector develops symptoms. Conclusions: High infectiousness immediately prior to symptom onset highlights the importance of continued contact tracing until effective vaccines have been distributed widely, even if contacts from a short time window before symptom onset alone are traced. Funding: Engineering and Physical Sciences Research Council (EPSRC).
Full-text available
In this contribution, some textual portions of the Leonardo da Vinci’s work were analyzed with the aim to highlight how, moving from Aristotle and going beyond him, he combines the intermediate positions that, from the Greek philosopher, passing through Buridan, arrive to Newton. This has been performed following a path that passes through the formulation of the principle of causality, the use of the concept of linear relationship (pyramidal law) between cause and effect and the introduction of a duration of the impression (memory) of mechanical systems. In the framework of the studies aimed to a valorization of Leonardo as a scientist, which is a crucial aspect in the analysis of the Leonardo genius, the present work sheds a new light on his intuitions about some fundamental physics concepts as well as about the conceptual model that, several centuries later, will be formalized in the modern linear response theory.
Full-text available
The introduction of fractional-order derivatives to epidemiological compartment models, such as SIR models, has attracted much attention. When this introduction is done in an ad hoc manner, it is difficult to reconcile parameters in the resulting fractional-order equations with the dynamics of individuals. This issue is circumvented by deriving fractional-order models from an underlying stochastic process. Here, we derive a fractional-order infectivity and recovery Susceptible Infectious Recovered (SIR) model from the stochastic process of a continuous-time random walk (CTRW) that incorporates a time-since-infection dependence on both the infectivity and the recovery of the population. By considering a power-law dependence in the infectivity and recovery, fractional-order derivatives appear in the generalised master equations that govern the evolution of the SIR populations. Under the appropriate limits, this fractional-order infectivity and recovery model reduces to both the standard SIR model and the fractional recovery SIR model.
Fractional calculus is undergoing rapid and ongoing development. We can already recognize, that within its framework new concepts and strategies emerge, which lead to new challenging insights and surprising correlations between different branches of physics. This book is an invitation both to the interested student and the professional researcher. It presents a thorough introduction to the basics of fractional calculus and guides the reader directly to the current state-of-the-art physical interpretation. It is also devoted to the application of fractional calculus on physical problems, in the subjects of classical mechanics, friction, damping, oscillations, group theory, quantum mechanics, nuclear physics, and hadron spectroscopy up to quantum field theory. © 2011 by World Scientific Publishing Co. Pte. Ltd. All rights reserved.